Mr. Morgan on Survivorships. 
ft annuity equal to the interest of the given sum during the life 
“ of C after A, provided A should die before B. " The first 
of these is E, and if an annuity of £ 1. by prob. I, be denoted 
by Q, the second will be = The required value 
therefore will be = E — x O If the three lives he 
equal, the general theorem will become = — : x V__ CC 
C CCC, which may be derived from either of the fore- 
going rules, or from the different series given above. 
PROBLEM IV. 
To find the value of a given sum S, payable on the death 
and C, should B die before one life in particular (A). 
solution. 
The payment of S in the first yeardepends on the contingency 
of the three lives having become extinct (A having survived 
B), which is expressed by - ' a , and therefore the va- 
lue of S in this year will be = yCbc — me — bd + ma'. 
In the second and following years the sum S will become pay- 
able if either of five events should take place, ist. If the 
' three lives should drop in the year (B having died before A). 
2dly, If C should die in any of the preceding years, and A 
die after B in that particular year. gdly. If B and C 
should die in any of the preceding years, and only A die in 
that year. 4thly, If B should die in any of the preceding 
